We propose here a simple black-hole analog in vehicular-traffic dynamics. The corresponding causal diagram is determined by the propagation of the tail light flashes emitted by a convoy of cars on a highway. In addition to being a new black-hole analog, this illustrates how causal diagrams, so common in general relativity, may be useful in areas as unexpected as vehicular-traffic dynamics.

## I. INTRODUCTION

The area of analog models of gravitational systems has a splendid history. Its genesis can be traced back to the beginning of the 1980s with two independent papers: one by Vincent Moncrief^{1} and the other by Bill Unruh.^{2} Moncrief considered a perturbed fluid in a general relativistic spacetime and concluded that *one can study the stability of spherical accretion onto nonrotating stars by the methods used for black holes*, while Unruh^{2} showed that the perturbation of usual nonrelativistic irrotational fluids develops sonic horizons, which emit sound waves with a thermal spectrum analogous to the radiation derived by Hawking for usual black holes.^{3} After that, various analog black holes were discovered in Bose–Einstein condensates, shallow water waves, slow light, and so on (see Refs. 4 and 5 for reviews of analog models of gravity). Our vehicular-traffic black hole adds a new and appealingly simple model to this list.

We propose a simple black-hole analog in vehicular-traffic dynamics. The corresponding causal diagram is determined by the propagation of the tail light flashes emitted by a convoy of cars on a straight highway. Similar to a real black hole, our vehicular-traffic black-hole model exhibits event and apparent horizons and a Planck region where the causal structure breaks down. This model also illustrates how (informational) causal diagrams, so common in general relativity, may be useful in areas as unexpected as vehicular-traffic dynamics. It is possible that this simple physical-informational approach, complemented by more traditional ones,^{6} could help to avoid traffic accidents under impaired visibility conditions. If this occurs, it would constitute a quite curious relativity spin-off.

This paper is organized as follows: In Sec. II, we present our vehicular-traffic model. In Sec. III, we present the information-flow metric associated with the propagation of the brake-light signal in the simplified time-independent (stationary) regime. In Sec. IV, we show how the drivers' reaction time can give rise to a vehicular-traffic black hole. In Sec. V, we show that taking into account accidents on the track drives us to the nonstationary regime. In Sec. VI, we work out the more realistic nonstationary regime. In Sec. VII, we summarize our results and add some final remarks.

## II. THE VEHICULAR-TRAFFIC MODEL

Consider a caravan of equally spaced inertial cars initially moving (to the left) with velocity $ v j = const , \u2009 j = 1 , 2 , \u2026$ on a straight highway covered with Cartesian coordinates. The vehicles are assumed to be smaller than any other distance scale in the problem. We also assume that the brake lights can only be seen by adjacent vehicles and that, once the car ahead brakes, the car behind should also brake after some reaction time *T *>* *0. *T* will vary depending on the weather conditions. The more fog is present, the larger *T* will be. We are interested in following the propagation of the tail light flashes along the highway when the brakes are applied in succession (see Fig. 1).

*j*-th car at the instant

*t*. The propagation of the tail light flashes can be represented by the sequence

*j*-th vehicle starts to brake.

## III. THE STATIONARY REGIME

Let us begin by assuming the braking is smooth and brief enough such that $ v j = v \u2248 const$. Hence, the initial separation between contiguous vehicles $ | r j + 1 ( t ) \u2212 r j ( t ) | \u2261 D$ remains (approximately) the same even after the brakes are applied, avoiding shocks. In addition, we start with the assumption that $ T = T ( r )$ is simply a monotonic function of the position where the driver receives the tail light signal from the car ahead. Later on, we will remove this restriction and envisage the more realistic case where the visibility conditions vary also in time $ T = T ( t , r )$.

*v*on a preferred reference frame $S$, namely, the highway. The speed of the information propagation with respect to the vehicular medium is

^{7,8}Now, let us note that, since all the velocity scales in the problem are much smaller than the speed of light, we can assume Newtonian physics all around in our calculations. Thus, we use the usual nonrelativistic velocity composition to express the information-propagation velocity with respect to the highway $S$

*v*<

*0 in Eq. (6) in accordance with Fig. 1. Note that we are only interested in the “outgoing” information flow, i.e., from the*

*j*-th to the

*j*+

*1-th car.*

*dl*

^{2}. In the relativistic realm, neither space nor time has an absolute meaning on its own right. Rather, only a combination of both of them does. The type of combination depends on the case. Far from matter, singularities and other oddities, where inertial observers experience a homogeneous and isotropic space, and the line element is the Minkowski one (of special relativity)

*v*=

*0, Eq. (6) simply reads $ d s V T 2 = \u2212 c s 2 d t 2 + d r 2$ corresponding to the Minkowski line element (with*

*c*playing the role of the speed of light) but for $ | v | \u2260 0$, nontrivial effects may come out because the right-moving information is dragged by the “left-moving vehicular-traffic spacetime.”

_{s}*r*coordinates). Thus, the smaller the

*r*coordinate the larger is the reaction time

*T*(

*r*). Now, let us consider that, as we move from far east to the west, there exists a point,

*r*=

*r*, where

_{H}*T*(

*r*) is sufficiently large such that

*r*=

*r*fixes the location of an effective

_{H}*apparent horizon*like that of a black hole: for $ r < r H$ originally outgoing information lines will propagate “inwards” with respect to the highway (although they will always propagate “outwards” with respect to the convoy). This is so because the information propagates so slowly in comparison to the cars motion that, by the time that car

*j*+

*1 reacts to the signal, it has already passed the point where the car*

*j*emitted the signal. This will preclude the tail light brake signals from “properly” propagating through the convoy, thus causing accidents.

In the stationary regime, where $ T = T ( r )$ (*T* does not depend on the coordinate *t*), the apparent horizon will coincide with the *event horizon*, which is the black-hole frontier from inside which no information escapes. In general relativity, light rays emitted outwards at the event horizon of a black hole get frozen there.^{9}

At this point, one should notice that the existence of a vehicular-traffic black hole will only depend on a rapid enough deterioration of the visibility conditions as *r* decreases (see Fig. 1). In this case, the function $ D / T ( r )$ will decrease rapidly as *r* decreases, leading $ D / T ( r ) \u2192 | v |$ at some *r* = *r _{H}*. Ideally, a realistic

*T*(

*r*) function would demand that data will be experimentally collected on the ground, but this is much beyond the scope of this work. Nevertheless, rather than choosing an arbitrary reaction time

*T*(

*r*) to create a vehicular-traffic horizon and explore its nontrivial consequences, we show in Sec. IV how

*T*(

*r*) can be chosen to formally mimic the simplest black hole in nature: the Schwarzschild black hole, which is completely characterized by its mass

*M*. Readers who are not interested in this connection should just employ the

*T*(

*r*) defined in Eq. (11), refer to Fig. 2, and skip directly to Sec. V.

## IV. THE VEHICULAR-TRAFFIC BLACK HOLE

*M*. The line element of a Schwarzschild black hole (omitted the angular part) can be cast in advanced Eddington–Finkelstein (E-F) coordinates $ ( t \xaf , r )$ as

^{10,11}

*c*by a convenient analogue speed $ C = 200 \u2009 km / h$. The similarity between Eqs. (6) and (8) is unambiguous, but there is a relevant difference: While the latter is already all determined, the former depends on $ c s ( r )$, which varies according to the visibility conditions. Here, we will select

*T*(

*r*) that yields a “Schwarzschild” analog vehicular-traffic black hole.

*T*(

*r*) such that the right-hand sides of both of these equations are formally equal for

*T*(

*r*) is assumed to be arbitrary large in the interval $ 0 \u2264 r \u2264 r P$. Note that $ T ( r ) \u2192 \u221e$ as $ r \u2192 r H$. It is important to keep in mind that our information-flow analysis will be restricted to the domain $ r \u2265 0$, although the highway is assumed to be unbounded, $ r \u2208 ( \u2212 \u221e , + \u221e )$.

In Fig. 2, we superimpose the smooth gray lines of the black-hole radial outgoing null worldlines in $ ( t \xaf , r )$ coordinates (10) (with $ const = C t 1 * \u2212 r 1 * \u2212 2 r H \u2009 ln \u2009 | r 1 * / r H \u2212 1 |$) on the dotted lines of our brake-light lines in the usual (*t*, *r*) Cartesian coordinates (12). The reason why the *continuous but nonsmooth* dotted lines of the brake lights overlap so well with the *smooth* gray lines of the black-hole outgoing worldlines is because we are dwelling in the small-*D* regime: $ D / c s \xb7 | d c s / d r | \u226a 1$. (This is in agreement with our assumption that the caravan should be seen as a sort of “vehicular medium.”) The vertical broken line at *r* = *r _{H}* corresponds to the black-hole horizon, which means that the visibility conditions at $ r P < r < r H$ slow down drivers' reactions enough to make outgoing information lines propagate “inwards” with respect to the highway. The gray region, $ 0 \u2264 r \u2264 r P$, was excised from our information-flow graph since

*T*(

*r*) diverges at

*r*=

*r*: vehicles that get the front-car brake signal at $ 0 \u2264 r \u2264 r P$ do not react ever. In general, the brake-light line will terminate somewhere in $ r < r P$ due to cars that receive the front-car tail light flash at $ r \u2273 r P$, braking at $ r < r P$, eventually. In analogy to relativity, the region $ 0 \u2264 r \u2264 r P$ will play the role of an information Planck region, where the theory will be assumed to break down. It is commonly assumed that black hole singularities are mathematical artifacts stemming from the fact that general relativity does not comply with quantum rules. Whether or not this is the case, we do not know. Nevertheless, it is fair to say that we shall not trust the causal structure predicted by general relativity close enough to such extreme regions as given by the Planck scale $ G \u210f / c 3 \u223c 10 \u2212 33 \u2009 cm$.

_{P}^{12}

We must emphasize that despite all the similarities between Eqs. (6) and (8), they only concern the propagation of information. The vehicle worldlines are not bound to any light cones shown in Fig. 2. In the stationary regime in which we are dwelling so far, $ v j = v = const$ and nothing would stop the vehicles from crossing the *r *=* *0 axis. However, the appearance of singularities due to collisions begs us to go beyond the stationary regime, changing this state of affairs.

## V. VEHICULAR-TRAFFIC SINGULARITY

*a*, during the time interval

*a*>

*0.) Figure 3 depicts a situation where both cars,*

*j*and

*j*+

*1, apply the brakes, attain a reduced velocity*

*v*, and move on without colliding with each other. A direct comparison with Fig. 2 indicates this happens

_{r}*outside the hole*, $ r > r H$, since outgoing information flows outwards. In contrast, the snapshot in Fig. 4 happens

*inside the hole*at $ r P < r < r H$, since outgoing information flows inwards. Depending on the physical situation, the cars still may not collide, but generally they will. There are different possibilities the

*j*+

*1-th car behind hits the*

*j*-th car ahead depending on whether or not the front car secures

*v*and the back car starts braking. In Fig. 4, the

_{r}*j*-th car reacts fast enough to begin braking but not to avoid a collision with the

*j*−1-th car, which attained the speed

*v*.

_{r}The first collision of a vehicular pileup will be interpreted as the appearance of a singularity since the rest of the convoy will collide in succession at the same spot. The fact that the singularity forms at some given time, rather than having existed forever, demands we abandon our initial stationarity assumption. Hence, let us move on and consider the physical situation where $ T = T ( t , r )$.

## VI. NONSTATIONARY REGIME

*T*(

*t*,

*r*) is assumed to be arbitrary large inside the Planck region. As before, the boundary of the Planck region is defined by the line where

*T*(

*t*,

*r*) diverges (see Fig. 5). Note that

*T*(

*t*,

*r*) reproduces Eq. (11) for $ t > t f$. In the region $ t i \u2264 t \u2264 t f$, the

*T*(

*t*,

*r*) function is designed to connect continuously the previous asymptotic regions.

In Fig. 5, we exhibit the information flow diagram for the nonstationary case. The propagation of the brake-light signals is depicted accordingly by dotted lines. For $ t < t i$, the information flow is described by homogeneously distributed straight dotted lines, corresponding to an effective Minkowski spacetime. However, as the weather conditions deteriorate, starting at *t* = *t _{i}*, the drivers react slower and an event horizon appears (see arrow), growing up to

*r*=

*r*at

_{H}*t*=

*t*. The information propagation lines coincide with the ones shown in Fig. 2 for $ t > t f$. Inside the event horizon, the information lines cannot go far away, in contrast to the ones outside the event horizon. The event horizon contains the apparent horizon, which emerges at $ t A H = ( t i + t f ) / 2$ and increases until it merges with the event horizon at

_{f}*t*=

*t*. Inside the apparent horizon, outgoing information flows inwards. The boundary of the Planck region is denoted by the internal dashed line, which equals

_{f}*r*=

*r*in the stationary region, $ t > t f$. The spots where the vehicles pile up lie inside the Planck region (or beyond:

_{P}*r*<

*0) in Fig. 5.*

## VII. SUMMARY AND FINAL REMARKS

In summary, let us assume a convoy of vehicles moving to the left, as in Fig. 1, and the first vehicle begins to brake at *r *=* *0 before the formation of the event horizon (below the arrow in Fig. 5). In this case, all the following cars have time to react properly, secure the reduced speed $ | v r |$, and pass safely through the foggy region. On the other hand, if the first vehicle applies the brakes after the formation of the event horizon (above the arrow), the future development will be entirely different. The outgoing information may still flow outwards for awhile depending on whether the brakes are applied before the formation of the apparent horizon, but, eventually, will flow inwards as it enters the apparent horizon, leading to accidents. The first condemned car of the convoy is represented by the small disk, which will be rammed by the one behind it. Some other pairwise shocks will occur on the track further but the vehicle responsible for the massive accident will be the one that receives the brake signal from the car ahead (large disk) inside the Planck region. Inside this region, the visibility conditions are so impaired the driver does not react to the brake signal, ramming the car straight ahead (see the target mark in Fig. 5). This is the spot where the rest of the convoy will inadvertently pile up.

As a result, accidents could be avoided if the drivers were warned to diminish the velocity before entering the event horizon. For this purpose, templates of risky areas should be prepared in advance to forecast where the event horizon would be formed as the visibility conditions deteriorate.

## ACKNOWLEDGMENTS

L.K.S. and G.E.A.M. were fully and partially supported by the São Paulo Research Foundation (FAPESP) under Grant No. 2019/18616-4 and Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) under Grant No. 301544/2018-2.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

## REFERENCES

*General Relativity*